Mihai D. Staic and Vladimir Turaev- Remarks on 2-dimensional HQFT’s

8/3/2019 Mihai D. Staic and Vladimir Turaev- Remarks on 2-dimensional HQFTs

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arXiv:0912

.1380v2

[math.GT

]2Apr2010

Remarks on 2-dimensional HQFTs

Mihai D. Staic and Vladimir Turaev

Abstract. We introduce and study algebraic structures underlying 2-dimen-

sional Homotopy Quantum Field Theories (HQFTs) with arbitrary targetspaces. These algebraic structures are formalized in the notion of a twistedFrobenius algebra. Our work generalizes results of Brightwell, Turner, and the

second author on 2-dimensional HQFTs with simply-connected or asphericaltargets.

Introduction

A fruitful idea in topology is to construct invariants of manifolds that behavefunctorially with respect to gluings of manifolds along the boundary. More for-mally, one associates to every closed oriented d-dimensional manifold M a finitedimensional vector space VM and to every compact oriented (d + 1)-dimensionalcobordism (W,M,N) a homomorphism W : VM VN. The conditions satis-fied by (V, ) in order to obtain a consistent theory are formalized in the notion

of a (d + 1)-dimensional Topological Quantum Field Theory (TQFT), see [At].The second author introduced more general Homotopy Quantum Field Theories(HQFTs), see [T1]. One can think of a (d + 1)-dimensional HQFT as of a TQFTfor d-dimensional manifolds and (d+1)-dimensional cobordisms endowed with mapsto a fixed space X.

The 1-dimensional HQFTs with target X are easily classified in terms of fi-nite dimensional representations of 1(X). A study of 2-dimensional HQFTs ismore involved and leads to interesting algebra. For contractible X, the cate-gory of 2-dimensional HQFTs with target X is equivalent to the category of 2-dimensional TQFTs and is known to be equivalent to the category of commutativefinite-dimensional Frobenius algebras. If X = K(G, 1) is an Eilenberg-MacLanespace determined by a group G, then the category of 2-dimensional HQFTs withtarget X is equivalent to the category of so-called crossed Frobenius G-algebras,

see [T1]. IfX = K(A, 2) is an Eilenberg-MacLane space determined by an abeliangroup A, then the category of 2-dimensional HQFTs with target X is equivalent tothe category of Frobenius A-algebras, see [BT].

Key words and phrases. Topological Quantum Field Theory, Frobenius algebra, k-invariant.The work of M. Staic was partially supported by the CNCSIS project Hopf algebras, cyclic

homology and monoidal categories contract nr. 560/2009. The work of V. Turaev was partially

supported by the NSF grants DMS-0707078 and DMS-0904262.

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2 MIHAI D. STAIC AND VLADIMIR TURAEV

In this paper we address the case where both groups G = 1(X) and A = 2(X)are allowed to be nontrivial. An important role will be played by the first k-invariantk H3(G, A) of X. We shall introduce and study algebraic structures underlying2-dimensional HQFTs with target X. We call them twisted Frobenius algebras or,shorter, TF-algebras over (G,A,k). Briefly speaking, a TF-algebra is a G-gradedalgebra of A-modules which is associative up to a 3-cocycle representing k andwhich satisfies a form of commutativity. This work is a step towards classificationof 2-dimensional HQFTs with target X (for a different approach, see [PT]).

The paper is organized as follows. We introduce TF-algebras in Section 1. InSection 2 we recall the notion of an HQFT and the definition of the first k-invariantof a topological space. In Section 3 we derive from any 2-dimensional HQFT theunderlying TF-algebra. In Section 4 we compute the underlying TF-algebras of thecohomological 2-dimensional HQFTs.

Throughout the paper the symbol K denotes a field and = K.

1. Twisted Frobenius Algebras

1.1. Preliminaries. In this section G is a group with neutral element andA is a left G-module with neutral element 1 = 1A. We use multiplicative notationfor the group operation in A. As usual, the group ring of A with coefficients in Kis denoted K[A]. The action of G on a A is denoted by a.

To calculate the cohomology H(G, A), we use the standard cochain complex

C0(G, A)0

C1(G, A)1

C2(G, A)2

C3(G, A)3

C4(G, A)

where Cn(G, A) = M ap(Gn, A) for any n 0. For small n, the coboundaryhomomorphism n : Cn(G, A) Cn+1(G, A) is given by the following formulas:

0(a)() = a a1

for any a A = C0(G, A) and G,

1()(, ) = () ()1()

for any C1(G, A) and , G,

2()(, , ) = (, ) (,)1(,) (, )1

for any C2(G, A) and , , G,

3()(, , , ) = ( , , ) (,,)1(,,) (,,)1(, , )

for any C3(G, A) and , , , G.

1.2. Definition of TF-algebras. Fix from now on a normalized 3-cocycle : G3 A. Thus, for all , , , G,

( , , ) (,,)1 (,,) (,,)1 (, , ) = 1.

The word normalized means that for all , G,(,,) = (,,) = (, , ) = 1.

A twisted Frobenius algebra (TF-algebra) over the triple (G,A,) is a G-gradedK[A]-module V = GV such that

1) (Underlying module and action of A.) The underlying K-vector space of Vis finite-dimensional and for all G, u V, and a A,

au = au.(1.1)


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REMARKS ON 2-DIMENSIONAL HQFTS 3

2) (Multiplication.) We have a K-bilinear multiplication V V V carryingV V to V for all , G. For any u V, v V , w V with , , G,

(uv)w = (, , ) u(vw).(1.2)There is a unit element 1V V such that 1Vu = u1V = u for all u V.

3) (Inner product.) We have a non-degenerate symmetric K-bilinear form

: V V K

such that for all G, the pairing

V V1 K, u v (uv 1V)(1.3)

(where u V, v V1) is non-degenerate.4) (Projective action of G.) For each G, we have a K-linear isomorphism

: V V carrying V to V1 for all and such that

(av) =a (v) for any a A and v V,(1.4)

|V = idV : V V ,(1.5)

(1V) = 1V,(1.6)

vu = (u)v for any u V, v V ,(1.7)

((u) (v)) = (u v) for any u, v V,(1.8)

(u) (v) = l, (uv) for any , G and u V, v V ,(1.9)

where

l, = (1,1, ) (1, , )1 ( , , ) A,

|V = h

, ( )|V for all , , G,(1.10)where

h, = ( ( )1, , ) (,1, )1( , , ) A.

5) (The trace condition.) For any , G and c V11 ,

Tr((11, , ) c : V V)

= Tr((11,1, ) 1 c : V V),(1.11)

where c is left multiplication V V, v cv by c and Tr is the standard trace oflinear maps.

We have the following elementary consequences of the definition. The K[A]-bilinearity of multiplication in V implies that

a(uv) = (au)v = u(av)(1.12)for all a A and u, v V. Note that if u V and v V , then for all a A,

a(uv) = (au)v = (au)v = a(uv)

and similarly a(uv) = a(uv). Formula (1.10) applied to = = implies that

= id : V V.(1.13)

In the following lemma and in the sequel the pairing (1.3) is denoted by .


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Lemma 1.1. For any a A, u V, v V1 with G,

(au v) = (u av),

(u v) = 1((1, , 1)1 v u).

For any , G, u V, v V1 ,

1((u) (v)) = (l

,1u v).

For any , G, u V, v V and w V()1 ,

(uv w) = ((,, ()1) u vw).

Proof. We check only the first two identities leaving the other two to thereader. We have

(au v) = ((au)v 1V)

= (u(av) 1V)

= (u av).

Formulas (1.5), (1.13), and (1.10) with = , = 1 imply the identity (v) =(1, , 1)1 v for all v V1 . Therefore

(u v) = (uv 1V)

= ((v)u 1V)

= ((1, , 1)1 vu 1V)

= 1((1, , 1)1 v u).

Given z K and a TF-algebra, we can multiply the inner product by z(keeping the rest of the data) and obtain thus a new TF-algebra. This operationon TF-algebras is called z-rescaling.

Let V, W be TF-algebras over (G,A,). A K[A]-isomorphism f : V Wis an isomorphism of TF-algebras if f is an isomorphism of algebras such that(f(u) f(v)) = (u v) for all u, v V and f = f for all G.

1.3. Examples. The definition of a TF-algebra over (G,A,) generalizes boththe notion of a crossed Frobenius G-algebra [T1] and the notion of a A-Frobenius

algebra [BT]. Consequently we have the following two sources of examples.

Example 1.2. If L is a crossed Frobenius G-algebra, then L is a TF-algebra over(G,A,) where A is the trivial group and is the trivial cocycle.

Example 1.3. IfV is a A-Frobenius algebra, then V is a TF-algebra over (G,A,)where G is the trivial group and is the trivial cocycle.

Further examples of TF-algebras are constructed in Section 4.


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1.4. Coboundary equivalence. Let : G3 A be a normalized 3-cocycle.Given a normalized 2-cochain : G2 A, the map = 2() : G3 A is anormalized 3-cocycle cohomological to . Using , we can transform a TF-algebraV over (G,A,) into a TF-algebra V over (G,A,) as follows. The underlyingG-graded K[A]-modules ofV and V are the same. The inner product on V = Vis the same as in V. Multiplication on V is defined by

u v = (, )1 u v,

for u V, v V , where is multiplication in V. Given G, the automorphism of V

is defined by

|V = (, )1 (1, ) |V

for all G. Direct computations show that V is a TF-algebra over (G,A,).We say that V is obtained from V by a coboundary transformation. This transfor-

mation defines an equivalence between the category of TF-algebras over ( G,A,)and their isomorphisms and the category of TF-algebras over (G,A,) and theirisomorphisms.

Given k H3(G, A), the coboundary transformations and the isomorphismsgenerate an equivalence relation on the class of TF-algebras over the triples (G,A,)where : G3 A runs over all normalized 3-cocycles representing k. This relationis called coboundary equivalence and the corresponding equivalence classes are calledTF-algebras over (G,A,k).

2. Preliminaries on HQFTs and k-invariants

2.1. HQFTs. We recall the definition of an HQFT from [T1], [T2]. We say

that a topological space is pointedif all its connected components are provided withbase points. By maps of pointed spaces we mean continuous maps carrying basepoints to base points.

Fix a pointed path connected topological space X. An X-manifold is a pair(M, g), where M is a pointed closed oriented manifold and g is a map M X. AnX-homeomorphism between (M, g) and (M, g) is an orientation preserving (andbase point preserving) homeomorphism f : M M such that g = gf. An emptyset is considered as an X-manifold of any given dimension.

An X-cobordism between X-manifolds (M0, g0) and (M1, g1) is an orientedcompact cobordism (W, M0, M1) endowed with a map g : W X such that W =(M0) M1 and g|Mi = gi for i = 1, 2. Note that W is not required to be pointed,but M0 and M1 are pointed. An X-homeomorphism between two X-cobordisms(W, M0, M1, g) and (W

, M0, M1, g

) is an orientation preserving homeomorphism

of triples F : (W, M0, M1) (W, M0, M1) such that g = gF and F restricts toX-homeomorphisms M0 M0 and M1 M

1. For example, for any X-manifold

(M, g) we have the cylinder cobordism (M [0, 1], M 0, M 1, g) between twocopies of (M, g). Here g is the composition of the projection M [0, 1] Mwith g. Any X-homeomorphism of X-manifolds multiplied by id[0,1] yields an X-homeomorphism of the corresponding cylinder cobordisms.

A (d+1)-dimensional Homotopy Quantum Field Theory (V, ) with target Xassigns a finite-dimensional K-vector space VM to any d-dimensional X-manifold,


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a K-isomorphism f# : VM VM to any X-homeomorphism of d-dimensional X-manifolds f : M M and a K-homomorphism (W) : VM0 VM1 to any (d +1)-dimensional X-cobordism (W, M

0, M

1). These vector spaces and homomorphisms

should satisfy the following axioms:(1) For any X-homeomorphisms of d-dimensional X-manifolds f : M M,

f : M M, we have (ff)# = f#f#.(2) For any disjoint d-dimensional X-manifolds M, N, there is a natural iso-

morphism VMN = VM VN.(3) V = K.(4) For any X-homeomorphism of (d + 1)-dimensional X-cobordisms

F : (W, M0, M1, g) (W, M0, M

1, g

),

the following diagram is commutative:

V(M0,g|M0)(F|M0)# V(M0,g|M

0)

(W,g)

(W,g)

V(M1,g|M1)(F|M1)# V(M1,g|M

1)

(5) If a (d+1)-dimensional X-cobordism W is a disjoint union ofX-cobordismsW1, W2, then (W) = (W1) (W2).

(6) If an X-cobordism (W, M0, M1) is obtained from two (d+1)-dimensional X-cobordisms (W0, M0, N) and (W1, N

, M1) by gluing along an X-homeomorphismf : N N, then

(W) = (W1)f#(W0) : VM0 VM1 .

(7) For any d-dimensional X-manifold (M, g),

(M [0, 1], M 0, M 1, g) = id : VM VM,

where we identify M 0 and M 1 with M in the obvious way and the triple(M [0, 1], M 0, M 1, g) is the cylinder cobordism over (M, g).

(8) For any (d + 1)-dimensional X-cobordism (W, M0, M1, g), the homomor-phism (W) is preserved under homotopies of g constant on W = M0 M1.

Remark 2.1. In this definition we follow [T2] rather than [T1]. The difference isthat axiom (7) in [T2] is weakened in comparison with the corresponding axiom in[T1].

For shortness, HQFTs with target X will be also called X-HQFTs. For ex-amples of X-HQFTs, see [T1], [T2]. Note here that any cohomology class Hd+1(X; K) determines a (d + 1)-dimensional X-HQFT (V, ).

An isomorphism of X-HQFTs : (V, ) (V, ) is a family of K-isomor-phisms {M : VM VM}M, where M runs over all d-dimensional X-manifolds, = idK , MN = M N for all M, N, and the natural square diagrams associ-ated with the X-homeomorphisms and X-cobordisms are commutative.

2.2. The k-invariant. Let X be a path connected topological space with basepoint x0. We recall from [EM] the definition of the first k-invariant of X.

Let p : [0, 1] S1 = {z C | |z| = 1} be the map carrying t [0, 1] toi exp(2it). We provide S1 with clockwise orientation and base point i =

p(0) = p(1). Set G = 1(X, x0) and recall that A = 2(X, x0) is a left G-module


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in the standard way. For each G, fix a loop u : S1 X carrying i to x0and representing . We assume that the loop u representing the neutral element G is the constant path at x

0. Fix a 2-simplex

2with vertices v

0, v

1, v

2. For

every , G, fix a map f, : 2 X such that for all t [0, 1],

f,((1 t)v0 + tv1) = up(t), f,((1 t)v1 + tv2) = u p(t),

f,((1 t)v0 + tv2) = u p(t) .

We assume that f, is the constant map 2 {x0} and that for all G, themaps f, and f, are the standard constant homotopies between two copiesof u. We call the collection = {{u}G; {f,},G} a basic system of loopsand triangles in X.

Figures 114 below represent X-surfaces using the following conventions. Thevertices in each figure are labeled by integers; the vertex labeled by an integer i isdenoted vi. With every edge in the figure we associate an element of G called itslabel. For some edges, the labels are indicated by Greek letters in the picture. The

labels of all the other edges can be computed uniquely using the following rule: inany triangle vivjvk with i < j < k the label of the edge vivk is the product of thelabels of vivj and vjvk. Each figure below represents a compact oriented surface endowed with a map f : X. The map f carries all vertices of to the basepoint x0 X. The restriction of f to any edge vivj with i < j and with label Gis given by f((1 t)vi + tvj) = up(t) for all t [0, 1]. The restriction of f to anytriangle vivjvk with i < j < k is given by

f(t0vi + t1vj + t2vk) = f,(t0v0 + t1v1 + t2v2),

where , are the labels of vivj and vjvk respectively and t0, t1, t2 run over non-negative real numbers whose sum is equal to 1. The map f : X is consideredup to homotopy constant on .

Figure 1. A map 3 X

Let 3 be the standard 3-simplex with vertices v0, v1, v2, v3. Given , , G,

Figure 1 describes a map from the 2-sphere 3 to X. Here the labels of the edgesv0v2, v1v3, and v0v3 are ,, and , respectively. We take v0 as the base pointof 3. With this choice, the map 3 X in Figure 1 represents an elementof A = 2(X, x0) denoted (, , ). This defines a map =

: G3 A. It isknown that is a cocycle. It follows from the definitions (and from the choice of themaps f,, f,) that is normalized in the sense of Section 1.1. The cohomologyclass of in H3(G, A) is independent of the choice of . This cohomology class isthe first k-invariant of X.


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3. The TF-algebra underlying a 2-dimensional HQFT

Let X be a path connected topological space with base point x0. We derivefrom any 2-dimensional X-HQFT (V, ) its underlying TF-algebra.

For shortness, one-dimensional X-manifolds will be called X-curves and two-dimensional X-cobordisms will be called X-surfaces. We say that two X-surfaceswith the same bases

(W, M0, M1, g : W X) and (W, M0, M1, g

: W X)

are h-equivalent if there is a homeomorphism F : W W constant on the bound-ary and such that gF : W X is homotopic to g via a homotopy constant on theboundary. The axioms of an HQFT imply that then (W) = (W) : VM0 VM1 .

Fix a basic system of loops and triangles = {{u}G; {f,},G} in X,where G = 1(X, x0). Let : G

3 A = 2(X, x0) be the cocycle constructed inSection 2.2. We construct a TF-algebra V over (G,A,) in several steps.

Step 1 (Underlying module and action of A). For each G, consider thepointed oriented circle S1 and the map u : S

1 X as in Section 2.2. The pair(S1, u) is an X-curve. Set V = V(S1,u). By the definition of an HQFT, V is afinite-dimensional K-vector space.

For all G, the group A = 2(X, x0) acts on V by

av = (S(, a))(v).

where a A, v V, and S(, a) is the X-annulus obtained as a connected sum ofthe cylinder cobordism over (S1, u) and a map S

2 X representing a. The X-annulus S(, a) is shown in the left part of Figure 2. Here and below we use externalarrows to indicate the edges glued to each other (keeping the surface oriented).

Observe that the X-annulus in the right part of Figure 2 is h-equivalent toS(, a). Therefore, av = av for all a A and v V.

Figure 2. The X-annulus S(, a)

Step 2 (Multiplication). For , G, consider the X-cobordism (a disk with

two holes) D(, ) in Figure 3. Composing the map

V V V V , (u, v) u v

with the homomorphism

(D(, )) : V V V

we obtain a K-linear multiplication V V V. This extends to multiplicationin G V by linearity.


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Figure 3. The X-disk with 2 holes D(, )

It is clear that for any a A, the glueing of S(, a) to D(, ) along the leftbottom base of D(, ) and the glueing of S(,a) to D(, ) along the top baseof D(, ) yield h-equivalent X-surfaces. Applying , we obtain (au)v = a(uv)

for any u V and v V . Similarly, (au)v = u(av). Hence, multiplication inG V is K[A]-bilinear.To check the twisted associativity (1.2), note that (uv)w is computed by ap-

plying to the X-surface (a disk with three holes) obtained by glueing D(, ) toD(,). Similarly, u(vw) is computed by applying to the X-surface obtainedby glueing D(, ) to D(,). These X-surfaces are shown in Figure 4 where theexternal arrows indicating the glueing of sides are omitted. It is easy to see thatthe left X-surface is h-equivalent to a connected sum of the right X-surface withthe map S2 X used to define (, , ) A. This implies Formula (1.2). Notefor future use that the narrow collar-type rectangles in Figure 4 play no role inthe argument but are necessary to define the X-surfaces at hand. In some of thefigures below we omit these collar rectangles to simplify the figures.

Figure 4. Proof of the identity (uv)w = (, , ) u(vw)

Next consider an oriented 2-disk B mapped to {x0} X and viewed as anX-cobordism between an empty set and (S1, u). We have a map (B) : K V,and we set 1V = (B)(1). One easily sees that 1Vu = u1V = u for all u V.

Step 3 (Inner product). Consider the X-annulus C(, ) shown in Figure 5.The subscript reflects the fact that the orientation on both boundary com-ponents of the annulus is opposite to the orientation induced from the annulus.Set

= (C(, )) : V V1 K.


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Figure 5. The X-annulus C(, )

The X-annulus C(, ) can be obtained by glueing the X-surfaces C(, ),D(, 1), and B. This yields

(u v) = (C

(, ))(u v) =

(uv 1V

)

for all u V and v V1 .The X-surfaces shown in Figure 6 are h-equivalent. Applying to the surface

on the left we get (idV1

)( idV1

), where is a homomorphism K V1 V. Applying to the surface on the right we get idV1 . The resultingequality (idV

1 )( idV

1) = idV

1implies the non-degeneracy of .

Figure 6. Proof of the non-degeneracy of

Note finally that the map C(, ) X used to define is the constant mapwith value x0. This easily implies that the form is symmetric.

Step 4 (Projective action of G). Consider the X-annulus C+(, ) shown inFigure 7. The subscript + reflects the fact that the orientation on the boundary

component corresponding to is opposite to the orientation induced from theannulus, while the orientation on the boundary component corresponding to 1

is induced from the orientation of the annulus. Set

= (C+(, )) : V V1 .

The identity (av) =a (v) for a A and v V follows from the fact

the X-annulus obtained by glueing C+(, ) to S(, a) is h-equivalent to the X-annulus obtained by glueing S(1, a) to C+(, ).


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Figure 7. The X-annulus C+(, )

Figure 8 represents two h-equivalent X-annuli (it is understood that the verticalsides of the right rectangle are glued in the usual way to make an annulus). The

left X-annulus is C+(, ). The right X-annulus is obtained by glueing C+(, )represented by the lower rectangle to an X-annulus C(, ) represented by theupper rectangle. This proves that is invertible and

1 = (C

(, )).

Figure 8. Proof of the invertibility of

To prove that |V = idV : V V , consider the X-annuli in Figure 9.Using the Dehn twist of an annulus about its core circle, one easily observes thatthese two X-annuli are h-equivalent. The X-annulus on the right is C+(, ), andthe associated map : V V is the identity by Axiom (7) of an HQFT. Itis easy to see that a connected sum of the left X-annulus with the map S2 Xused to define ( , , ) A is h-equivalent to C+(, ). Since ( , , ) = 1, weobtain idV = |V .

The equality (1V) = 1V follows from the fact that the X-disk obtained byglueing the X-disk B to the bottom of C+(, ) is h-equivalent to B.

To prove the identity (1.7), consider the X-surfaces (disks with two holes) inFigure 10 (we omit in the figure the collar rectangles glued to some external edges;

they play no role in the argument and the reader may easily recover them). Thefirst X-surface is obtained by glueing the X-annulus C(, ) to the right bottomboundary component (S1, u) of D(, ). Since (, , ) = 1, this X-surface ish-equivalent to the second X-surface in Figure 10. The third X-surface is obtainedfrom the second one by separating the left face along the -labeled edge v0v2 andgluing it on the right along the edges indicated by the external arrows. Thus,the second and third X-surfaces are h-homeomorphic. The map from the thirdX-surface to X can be easily computed because its restriction to the face v1v3v4


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Figure 9. Proof of the identity |V = idV

(respectively to v0v2v4) is the constant homotopy of to itself (respectively, of to itself). This X-surface is h-equivalent to D(1, ). Applying , we obtain

v1

(w) = wv for all v V and w V1 . This is an equivalent form of (1.7).

Figure 10. Proof of the identity v1 (w) = wv

The identity ((u) (v)) = (u v) for u, v V follows from the factthat the X-annulus obtained by the glueing ofC+(, )C+(, ) to the bottomof C(, ) is h-equivalent to C(, ) (see Figure 11).

To prove that(u) (v) = l

, (uv)

for all u V and v V , consider the X-disks with three holes W1, W2, W3, W4in Figure 12 (again we omit the collar rectangles glued to some external edges).Clearly, the homomorphism (W1) : VV V1 carries uv to (u) (v).The X-surface W2 differs from W1 by two adjacent triangles v0v1v5 and v0v1v5both representing f1,1 : 2 X. These two copies of f1,1 can-cel each other, and so W2 is h-equivalent to W1. Thus, (W2) = (W1). The


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from W3 by canceling two adjacent copies of the singular triangle f, . ThereforeW4 is h-equivalent to W3 and (W4) = (W3). It is clear that (W4) = |V.Combining these equalities, we obtain the required identity.

Figure 13. Proof of the identity (u) = h, ((u))

Step 5 (The trace condition). Consider the three X-surfaces (punctured tori)W1, W2, W3 in Figure 14. All three are X-cobordisms between (S

1, u) and , where = 11. Clearly,

(W1) = (,1, ) (W2) and (W3) = ( , , ) (W2).

Consider the X-surface obtained by glueing C+(, ) and D(, ). If we identifythe two copies of (S1, u) we obtain W1. Similarly for the X-surface obtained byglueing D(, ) and C(, ) we can identify the two copies of (S1, u) in order toobtain W3. This implies the equality

Tr(( , , )c : V V) = Tr((,1, )1 c : V V).

We summarize the results above in the following theorem.

Theorem 3.1. To every 2-dimensional X-HQFT (V, ) and to every basic systemof loops and triangles = ({u}G, {f,},G) inX, we associate a TF-algebraV = V() over the triple (G = 1(X), A = 2(X),

: G3 A).


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Figure 14. Proof of the trace condition

It is obvious that the construction of V is functorial with respect to isomor-phisms of X-HQFTs. We now show that V does not depend on at least up tocoboundary equivalence (see Section 1.4).

Lemma 3.2. The TF-algebra V considered up to coboundary equivalence does not

depend on the choice of .Proof. Let = ({u}G, {f

,},G) be another basic system of loops

and triangles in X. Assume first that u = u for all G. It is clear that

up to homotopy constant on 2 the map f, : 2 X is a connected sum of

f, : 2 X and a certain map (, ) : 2 X carrying 2 to x0. Assigning

to each pair (, ) G2 the element of A = 2(X, x0) represented by (, ), weobtain a 2-cochain : G2 A. This cochain is normalized because by the definitionof a basic system of loops and triangles, f, = f, and f

, = f, for all G.

A direct comparison of the X-surfaces used to define the 3-cocycles , : G3 Aassociated with , and the TF-algebras V, V

shows that = 2() and

V

is obtained from V by the coboundary transformation determined by , i.e.,V

= (V).

In the case u = u for some , we construct a third basic system of loops andtriangles in X as follows. For each G, fix a homotopy h : S1 [0, 1] X

between the loops u, u : S

1 X representing . (It is understood that h is aconstant map to x0.) Recall the map p : [0, 1] S1 from Section 2.2. The map

h = h (p id[0,1]) : [0, 1]2 X

is a homotopy between up and up. We construct a map f

h, :

2 X by

gluing h, h , h to the sides of the singular simplex f, : 2 X. The system


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h = ({u}G, {fh,},G) satisfies all conditions of a basic system of loops and

triangles in X except possibly one: the maps fh, and fh, are not necessarily the

standard constant homotopies between two copies of up. However, f

h

, andfh, are homotopic rel

2 to these constant homotopies, as easily follows from

the assumptions on f, and f, and the definition of fh. Therefore, deforming

if necessary the maps fh, and fh,, we can transform

h into a basic system

of loops and triangles in X. The associated 3-cocycle : G3 A is equal to .Moreover, the homomorphisms

{(h) : Vu Vu}G

form an isomorphism V V

in the category of TF-algebras over (G,A,). By

the argument above, the TF-algebra V

is obtained from V

by a coboundarytransformation. This completes the proof of the lemma.

The following theorem shows that the isomorphism class of a 2-dimensionalX-HQFT is entirely determined by the underlying TF-algebra.

Theorem 3.3. Two 2-dimensional X-HQFTs are isomorphic if and only if theirunderlying TF-algebras are coboundary equivalent.

Proof. We fix a basic system of loops and triangles in X. It is obviousthat an isomorphism of X-HQFTs : (V1, 1) (V2, 2) induces an isomorphismV1 V

2 of TF-algebras over (1(X), 2(X),

). Therefore, the underlying TF-algebras are coboundary equivalent.

Conversely, assume that we have two X-HQFTs (V1, 1) and (V2, 2) such thattheir underlying TF-algebras are coboundary equivalent. If we use the same basicsystem = ({u}1(X), {f,},1(X)) of loops and triangles in X, then the

two algebras in question are TF-algebras over the same triple (1(X), 2(X), ).

Since they are coboundary equivalent, there is an isomorphism : V1 V2 . Welift to an isomorphism of the HQFTs. For every 1(X), denote by therestriction of to (V1 ) = (V1)(S1,u). For every connected X-curve (M, gM), there

are 1(X), an X-annuli a = aM : [0, 1] S1 X, and an X-homeomorphism

h = hM : (S1, aM|1S1) (M, gM) such that

aM|0S1 = u : S1 X and aM([0, 1] {}) = x0,

where is the base point of S1 and x0 is the base point of X. Notice that isuniquely determined by (M, gM). We define M : (V1)M (V2)M by the formula

M = h#22(a)1(a)1h1#1 ,

where h#i : (Vi)(S1,a|1S1) (Vi)(M,gM) is the K-isomorphism associated to h by

the HQFT (Vi, i). Next we show that M does not depend on the choice of a andh. Indeed, take another pair a, h which satisfies the above properties. Consider a(respectively a) as an X-cobordism between (S1, u) and (S

1, a|1S1) (respectively

(S1, a|1S1)) and observe that h1h is an X-homeomorphism between the top bases

of these X-cobordisms. Let C be the X-cobordism obtained by glueing these twoalong h1h (the orientation in the second cobordism should be reversed). Then Cis a cobordism between two copies of (S1, u). Clearly, C is the trivial X-annuli (asin axiom (7) of an HQFT) up to h-equivalence and the action of some c 2(X).


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Hence i(C) : (Vi ) (V

i ) is multiplication by c. Since is an isomorphism of

TF-algebras, we have

1(C) = 2(C)or equivalently

1(a)1(h1h)#11(a) = 2(a)

1(h1h)#22(a).

This implies that

h#22(a)1(a)1h1#1 = h#22(a)1(a)

1h1#1.

So M does not depend on the choice of a and h. Denote by the family of K-isomorphisms {M : (V1)M (V2)M}M where M runs over all X-curves. We claimthat is an isomorphism of HQFTs.

For any X-homeomorphism f : M N, we have

f#2M = f#2(hM)#22(aM)1(aM)1(hM)

1#1

= (f hM)#22(aM)1(aM)1

(f hM)1#1f#1

= Nf#1 .

So, is natural with respect to X-homeomorphisms.Next, notice that any compact oriented X-surface splits along a finite family

of disjoint simple loops into a disjoint union of X-disks with at most two holes.Deforming if necessary the map from the surface to X, we can choose the splittingloops to be X-curves X-homeomorphic to (S1, u), for some 1(X). Moreover,we can make the cuts in such a way that each component adjacent to the boundaryis X-homeomorphic to some aM and all the other X-surfaces are of types

B, S(, a), D(, ), C+(, ), C(, ), C++(, ).(3.1)

We consider the X-surface aM(id h1M ) : [0, 1] M X which is an X-cobordism

between M

= (M, u h1M ) and (M, gM). Axiom (4) of an HQFT implies that for

i = 1, 2,

i(aM(id h1M )) = (hM)#ii(aM)(hM)

1#i

.

From this formula and the definition of , we have

M1(aM(id h1M )) = (hM)#22(aM)(hM)

1#1

= (hM)#22(aM)(hM)1#2

(hM)#2(hM)1#1

= 2(aM(id h1M ))M .

Therefore the natural square diagram associated to and aM(idh1M ) is commuta-

tive. Since is a morphism of TF-algebras, the natural square diagrams associatedwith the X-surfaces listed in (3.1) are commutative. Finally, since any X-surfacecan be obtained by glueing a finite collection of the above X-surfaces along X-homeomorphisms, we get that the natural square diagrams associated to all X-surfaces are commutative. We conclude that is an isomorphism of HQFTs.

We expect that any TF-algebra can be realized by a 2-dimensional HQFT whichif true, together with Theorem 3.3 would yield a complete algebraic characterizationof 2-dimensional HQFTs with arbitrary targets.

Given a mapping of pointed topological spaces f : Y X, we can pull back a2-dimensional X-HQFT along f to obtain a 2-dimensional Y-HQFT. If f induces


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isomorphisms in 1 and 2, then the underlying TF-algebra of this Y-HQFT isisomorphic to the underlying TF-algebra of the original X-HQFT.

4. Simple TF-algebras and cohomological HQFTs

In this section we compute the underlying TF-algebras of the cohomological2-dimensional HQFTs. We begin by introducing a class of simple TF-algebras.

4.1. Simple TF-algebras and -pairs. Let G be a group with neutral ele-ment and A be a left G-module. Let : G3 A be a normalized 3-cocycle. ATF-algebra V = G V over (G,A,) is simple if dimK(V) = 1 for all G.We now classify simple TF-algebras in terms of so-called -pairs.

Form now on we endow the multiplicative abelian group K with the trivialaction of G. This allows us to apply notation of Section 1.1 to K-valued cochainson G. By a -pair, we mean a pair of maps g1 : G G K, g2 : A K such

that g2 is a Z[G]-homomorphism,

g1(, ) = g1(, ) = 1 for all G,(4.1)

and

2(g1) = g2 : G3 K.(4.2)

The Z[G]-linearity of g2 : A K means that g2 is a group homomorphism suchthat for all G, a A,

g2(a) = g2(a).(4.3)

For example, for any map : G K the pair

(g1 = 1() : G G K, g2 = 1 : A K)

is a -pair. We call it a coboundary -pair.

Lemma 4.1. Let(g1, g2) be a-pair. For each G, letV be the one-dimensionalvector space over K with basis vector l. We provide V = GV with a structureof an A-module by av = g2(a)v for all v V. We provide V with K-bilinearmultiplication by

ll = g1(, )1l(4.4)

for all, G. Let : V V K be the K-bilinear form such that

(l l) = g1(, ).(4.5)

Let : V V be the K-homomorphism defined by

(l) = g1(, )1g1(

1, ) l1 .(4.6)

for all, G. Then the G-graded vector space V with this data is a TF-algebraover (G,A,).

The TF-algebra constructed in this lemma is denoted by V(g1, g2).


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Proof. The K[A]-bilinearity of multiplication in V follows from the defini-tions. Formula (1.1) follows from the identity (4.3). It is sufficient to verify (1.2)for the basis vectors u = l, v = l, w = l:

(ll)l = (g1(, )1l) l

= g1(, )1g1(,)

1l

= g1(, )1g1(,)

1g2((, , )) l

= g2((, , )) l(l l)

= (, , ) l(l l).

Formula (4.1) implies that l is the unit element of V. The symmetry and non-degeneracy of are obvious. The non-degeneracy of the form : V V1 Kdefined by (1.3) follows from the formula (l l1) = g1(,

1)1. We check(1.4):

(al) = g1(, )1g1(

1, ) g2(a) l1

= g2(a) g1(, )

1g1(1, ) l1

= a (l).

Formulas (1.5) and (1.6) follow from the definitions. We check (1.7):

(l) l = g1(, )1g1(

1, ) l1 l

= g1(, )1 l = l l.

Similar computations prove (1.8) and (1.9). We now check (1.10). Observe that for, , G,

((l)) = g1(,1)1g1( ()

1, ) g1(, )1g1(

1, ) l()1

and

(l) = g1( , )1g1( ( )1, ) l()1.

Therefore (l) = h ((l)), where

h = g1(,)1g1( ( )

1, )

g1(,1) g1( ()

1, )1g1(, ) g1(1, )1.

Now, a direct computation using the definition of h, and the assumption that

g2 is a group homomorphism satisfying g2 = 2(g1) shows that g2(h, ) = h.Therefore

(l) = h ((l)) = h, ((l)).

To check the trace identity (1.11), observe that

g2((11, , )) = (2(g1))(

11, , )

= g1(, ) g1(1, )1g1(11, ) g1(11, )1.

and similarly

g2((11,1, )) = (2(g1))(

11,1, )

= g1(1, ) g1(, )

1g1(11, ) g1(

11,1)1,

It follows from the definitions that

Tr(c) = g1(, )1g1(

11, ) g1(11, 11)1


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and

Tr(1 c) = g1(1, )1g1(, ) g1(

11, )1.

Comparing these expressions, we obtain that

g2((11, , )) Tr(c) = g2((

11,1, )) Tr(1 c).

This formula is equivalent to (1.11).

Lemma 4.2. Any simple TF-algebraV = G V over(G,A,) such that(1V1V) = 1 is isomorphic to V(g1, g2) for a certain -pair (g1, g2). This -pair isdetermined byV uniquely up to multiplication ofg1 by

1() for a map : G K.

Proof. For each G, fix a nonzero vector l V. In the role of l Vwe take 1V . For any a A and G, we have al = g

2 (a)l with g

2 (a) K.

Clearly,

l = 1Al = (a1a) l = g2 (a1) g2 (a) l

and so g2 (a) K.

Given , G, we have ll = c(, ) l for some c(, ) K. Since thepairing (1.3) is non-degenerate, ll1 = 0. Thus, c(,

1) = 0 for all .We claim that g2 (a) does not depend on for every a A. Indeed,

(al) l1 = g2 (a) l l1 = g

2 (a) c(,

1) l

and

a(ll1) = ac(, 1) l = g

2(a)c(,

1) l.

Since (al) l1 = a(ll1) and c(, 1) = 0, we have g2 (a) = g

2(a). Set g2 =

g2 : A K. Since V is an A-module, the map g2 is a group homomorphism.Formula (1.1) implies the identity (4.3).

Given , G, we have

c(, 1) l = c(, 1) l l

= l(l l1)

= (, , 1)1(l l) l1

= (, , 1)1c(, ) l l1 .

Therefore c(, ) K. Let g1 : G G K be the map defined by g1(, ) =(c(, ))1 for all , . The identity ll = g1(, )

1l and Formula (1.2) yield

g1(, )1g1(,)

1l = g2((, , )) g1(,)1g1(, )

1l .

This implies that g2 = 2(g1). The equality (4.1) follows from the definitionsand the choice l = 1V . Thus, the maps g1, g2 form a -pair. It is clear thatV = V(g1, g2).

The second claim of the lemma follows from the fact that any two bases l ={l}G and l = {l}G in V as above are related by l = l

, where K

for all . The bases l, l yield the same map g2 : A K while the associatedmaps g1 : G G K differ by the coboundary of the map G K, .


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4.2. The group H2(G,A,; K). The -pairs (g1 : G G K, g2 : A K) form an abelian group H = H(G,A,) under pointwise multiplication. Theneutral element of H is the -pair (g

1= 1, g

2= 1). The inverse of a -pair (g

1, g

2)

in H is the -pair (g11 , g12 ). The coboundary -pairs form a subgroup of H.

Let H2(G,A,; K) be the quotient of H by the subgroup of coboundary -pairs.Lemma 4.2 yields a bijective correspondence between the set H2(G,A,; K) K

and the set of isomorphism classes of simple TF-algebras over (G,A,). Thiscorrespondence assigns to a pair (h H2(G,A,; K), z K) the isomorphismclass of the z-rescaled TF-algebra V(g1, g2), where (g1, g2) is an arbitrary -pairrepresenting h. The following theorem due to Eilenberg and MacLane computesH2(G,A,; K) in topological terms.

Theorem 4.3. ([EM]) Let X be a path connected topological space with base pointx0. LetG = 1(X, x0), A = 2(X, x0), and = {{u}G; {f,},G} be a basicsystem of loops and triangles in X. Let be a K-valued singular 2-cocycle on Xrepresenting a cohomology class [] H2(X, K). Then the pair (g1 : G G K, g2 : A K) defined by g1(, ) = (f,) for , G and g2(a) = (a) fora A is a -pair, where = : G3 A be the 3-cocycle defined in Section 2.2.Moreover, the formula [] (g1, g2) defines an isomorphism

H2(X, K) = H2(G,A,; K).

4.3. The underlying TF-algebras of cohomological HQFTs. We keepthe assumptions of Theorem 4.3.

Theorem 4.4. Let (V, ) be the 2-dimensional X-HQFT determined by H2(X, K) (see [T1]). If corresponds to (g1, g2) H

2(G,A,,K), then theunderlying TF-algebra of (V, ) is isomorphic to V(g1, g2).

Proof. Let V = G V = V be the underlying TF-algebra of (V

,

).The definition of V implies that V = V

(S1,u)

is a 1-dimensional K-vector space.

This vector space is generated by a vector p represented by the map p : [0, 1] S1

from Section 2.2 viewed as a fundamental cycle of S1. Multiplication in V iscomputed by

pp = (D(, ))(p p) = f

()(B)p ,

where , G, the map f : D(, ) X is determined by the structure of anX-surface in the disk with two holes D(, ), and B is a fundamental singular chainin D(, ) such that (B) = p p p . It is easy to see (cf. [T2]) that

g()(B) = (f,)1 = g1(, )

1.

So, pp = g1(, )1p . Similarly, for all a A = 2(X),

a p =

(S(, a))(p) = (a)p = g2(a)p.where (a) K is the evaluation of on a. Therefore V = V(g1, g2).

References

[At] M. Atiyah, Topological quantum field theory. Publications Mathematiques de lIHES, 68

(1988), 175186.[BT] M. Brightwell and P. Turner, Representations of the homotopy surface category of a simple

connected space. J. Knot Theory Ramifications 9 (2000), 855864.


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[EM] S. Eilenberg and S. MacLane, Determination of the second homology and cohomology groupsof a space by means of homotopy invariants. Proc. Nat. Ac. Sci. 32 (1946), 277280.

[PT] T. Porter and V. Turaev, Formal homotopy quantum field theories. I. Formal maps and

crossedC-algebras. J. Homotopy Relat. Struct. 3 (2008), 113159.[Ro] G. Rodrigues, Homotopy qunatum field theories and homotopy cobordism category in dimen-

sion 1+1. J. Knot Theory Ramifications 12 (2002), 287319.[T1] V. Turaev, Homotopy field theory in dimension 2 and group-algebras. math.QA/9910010

[T2] V. Turaev, Homotopy Quantum Field Theory. EMS Tracts Math. 10, European Math. Soc.Publ. House, Zurich, (2010).

Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN

47405, USA

Institute of Mathematics of the Romanian Academy, PO.BOX 1-764, RO-70700, Bu-

charest, Romania

E-mail address: [email protected]

Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN

47405, USA

E-mail address: [email protected]
http://arxiv.org/abs/math/9910010http://arxiv.org/abs/math/9910010

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Mihai D. Staic and Vladimir Turaev- Remarks on 2-dimensional HQFT’s